Doesn't look like that can be done at this point in time. Hopefully when the BETA program of Cortana ends, it will.
This is the reason why I've got at least eight other weather apps (those I've tested to be accurate for my location, not necessarily good in UI) pinned to my start screen.
I must mention, though, that what Cortana is "reporting" is different from what my Microsoft Weather app is reporting, so it seems that Cortana is sourcing its weather information from a different database, at least not exactly with what Microsoft's Weather app is using. It's just that tapping on the weather icon in Cortana launches Microsoft's weather app by default.